## Stability of Runge-Kutta methods for the generalized pantograph equation.(English)Zbl 0943.65091

Stability properties of Runge-Kutta (RK) methods applied to nonautonomous neutral dealy-differential equations with a constant delay of the form $u'(t)- Ku'(t-\tau)= e^t Lu(t)+ e^tMu(t-\tau),\quad t>0,\tag{$$*$$}$ are studied. Equation $$(*)$$ is obtained from the autonomous generalized pantograph equation $v'(t)- Kv'(\alpha t)= Lv(t)+ Mv(\alpha t),\quad 0<\alpha< 1,\tag{$$**$$}$ by the change of the independent variable $$u(t)=v(e^t)$$. When the RK matrix is regular, it is shown that the stability properties of the RK method for $$(*)$$ may be derived from those for a difference equation. Further, it is shown that some RK methods based on classical quadrature have a superior stability property with respect to $$(**)$$.

### MSC:

 65L20 Stability and convergence of numerical methods for ordinary differential equations 65L05 Numerical methods for initial value problems involving ordinary differential equations 34K40 Neutral functional-differential equations 65L06 Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations
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